#427244

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,14)(7,11,17,26)(8,12,20,29)(9,22,23,13)(15,27,18,24)(16,28,19,25)(21,34,33,30)(31,35,32,36);(5,7)(6,11)(8,19)(9,15)(10,17)(12,28)(13,24)(14,26)(16,20)(18,23)(21,32)(22,27)(25,29)(30,36)(31,33)(34,35);(7,15)(9,23)(11,24)(13,22)(16,31)(17,18)(19,32)(21,33)(25,35)(26,27)(28,36)(30,34);(2,4)(5,8)(6,12)(7,16,17,19)(9,21)(10,20)(11,25,26,28)(13,30)(14,29)(15,31,18,32)(22,34)(23,33)(24,35,27,36)

Polycyclic presentation:

2^9|a^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334440);